Fundamentals and Basic Terminology
1.10 SPECTRA
(a)
(c)
(e)
(b)
(d)
(f) p
p
p
t t t
Figure 1.7 Sound generation illustrated. (a) The piston moves right, compressing air as in (b). (c) The piston stops and reverses direction, moving left and decompressing air in front of the piston, as in (d).
(e) The piston moves cyclically back and forth, producing alternating compressions and rarefactions, as in (f). In all cases disturbances move to the right with the speed of sound.
p
p
p
t t
t
f f1 f
f1 f2 f3
Frequency bands (a)
(c)
(e)
(b)
(d)
(f) p2
p2
p2
Figure 1.8 Spectral analysis illustrated. (a) Disturbance p varies sinusoidally with time t at a single frequency f1, as in (b). (c) Disturbance p varies cyclically with time t as a combination of three sinusoidal disturbances of fixed relative amplitudes and phases; the associated spectrum has three single-frequency components f1, f2 and f3, as in (d). (e) Disturbance p varies erratically with time t, with a frequency band spectrum as in (f).
with the net result being a “wave” of positive and negative pressure transmitted along the tube.
If the piston moves with simple harmonic motion, a sine wave is produced; that is, at any instant the pressure distribution along the tube will have the form of a sine wave, or at any fixed point in the tube the pressure disturbance, displayed as a function of time, will have a sine wave appearance. Such a disturbance is characterised by a single frequency. The motion and corresponding spectrum are illustrated in Figure 1.8(a) and (b).
If the piston moves irregularly but cyclically, for example, so that it produces the waveform shown in Figure 1.8(c), the resulting sound field will consist of a combination of sinusoids of several frequencies. The spectral (or frequency) distribution of the energy in this particular sound wave is represented by the frequency spectrum of Figure 1.8(d). As the motion is cyclic, the spectrum consists of a set of discrete frequencies.
Although some sound sources have single-frequency components, most sound sources produce a very disordered and random waveform of pressure versus time, as
BN ' 10log10fC and fC ' fRfu (1.89a,b) illustrated in Figure 1.8(e). Such a wave has no periodic component, but by Fourier analysis it may be shown that the resulting waveform may be represented as a collection of waves of all frequencies. For a random type of wave the sound pressure squared in a band of frequencies is plotted as shown, for example, in the frequency spectrum of Figure 1.8(f).
Two special kinds of spectra are commonly referred to as white random noise and pink random noise. White random noise contains equal energy per hertz and thus has a constant spectral density level. Pink random noise contains equal energy per measurement band and thus has an octave or one-third octave band level that is constant with frequency.
1.10.1 Frequency Analysis
Frequency analysis is a process by which a time-varying signal is transformed into its frequency components. It can be used for quantification of a noise problem, as both criteria and proposed controls are frequency dependent. When tonal components are identified by frequency analysis, it may be advantageous to treat these somewhat differently than broad band noise. Frequency analysis serves the important function of determining the effects of control and it may aid, in some cases, in the identification of sources. Frequency analysis equipment and its use is discussed in Chapter 3.
To facilitate comparison of measurements between instruments, frequency analysis bands have been standardised. The International Standards Organisation has agreed upon “preferred” frequency bands for sound measurement and by agreement the octave band is the widest band for frequency analysis. The upper frequency limit of the octave band is approximately twice the lower frequency limit and each band is identified by its geometric mean called the band centre frequency.
When more detailed information about a noise is required, standardised one-third octave band analysis may be used. The preferred frequency bands for octave and one- third octave band analysis are summarised in Table 1.2.
Reference to the table shows that all information is associated with a band number, BN, listed in column one on the left. In turn the band number is related to the centre band frequencies, fC , of either the octaves or the one-third octaves listed in the columns two and three. The respective band limits are listed in columns four and five as the lower and upper frequency limits, fR and fu. These observations may be summarised as follows:
A clever manipulation has been used in the construction of Table 1.2. By small adjustments in the calculated values recorded in the table, it has been possible to arrange the one-third octave centre frequencies so that ten times their logarithms are the band numbers of column one on the left of the table. Consequently, as may be observed the one-third octave centre frequencies repeat every decade in the table. A practitioner will recognise the value of this simplification.
fu/fR ' 21/N N' 1,3 (1.90)
Table 1.2 Preferred frequency bands (Hz)
Band Octave band One-third octave band Band limits number centre frequency centre frequency Lower Upper
14 25 22 28
15 31.5 31.5 28 35
16 40 35 44
17 50 44 57
18 63 63 57 71
19 80 71 88
20 100 88 113
21 125 125 113 141
22 160 141 176
23 200 176 225
24 250 250 225 283
25 315 283 353
26 400 353 440
27 500 500 440 565
28 630 565 707
29 800 707 880
30 1,000 1,000 880 1,130
31 1,250 1,130 1,414
32 1,600 1,414 1,760
33 2,000 2 ,000 1,760 2,250
34 2,500 2,250 2,825
35 3,150 2,825 3,530
36 4,000 4,000 3,530 4,400
37 5,000 4,400 5,650
38 6,300 5,650 7,070
39 8,000 8,000 7,070 8,800
40 10,000 8,800 11,300
41 12,500 11,300 14,140
42 16,000 16,000 14,140 17,600
43 20,000 17,600 22,500
In the table, the frequency band limits have been defined so that the bands are approximately equal. The limits are functions of the analysis band number, BN, and the ratios of the upper to lower frequencies, and are given by:
where N = 1 for octave bands and N = 3 for one-third octave bands.
∆f ' fC 21/N&1
21/ 2N ' 0.2316fC for 1/3 octave bands
' 0.7071fC for octave bands
(1.91) The information provided thus far allows calculation of the bandwidth, ∆f of every band, using the following equation:
It will be found that the above equations give calculated numbers that are always close to those given in the table.
When logarithmic scales are used in plots, as will frequently be done in this book, it will be well to remember the one-third octave band centre frequencies. For example, the centre frequencies of the 1/3 octave bands between 12.5 Hz and 80 Hz inclusive, will lie respectively at 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 of the distance on the scale between 10 and 100. The latter two numbers in turn will lie at 1.0 and 2.0, respectively, on the same logarithmic scale.
Instruments are available that provide other forms of band analysis (see Section 3.12). However, they do not enjoy the advantage of standardisation so that the comparison of readings taken on such instruments may be difficult. One way to ameliorate the problem is to present such readings as mean levels per unit frequency.
Data presented in this way are referred to as spectral density levels as opposed to band levels. In this case the measured level is reduced by ten times the logarithm to the base ten of the bandwidth. For example, referring to Table 1.2, if the 500 Hz octave band which has a bandwidth of 354 Hz were presented in this way, the measured octave band level would be reduced by 10 log10 (354) = 25.5 dB to give an estimate of the spectral density level at 500 Hz.
The problem is not entirely alleviated, as the effective bandwidth will depend upon the sharpness of the filter cut-off, which is also not standardised. Generally, the bandwidth is taken as lying between the frequencies, on either side of the pass band, at which the signal is down 3 dB from the signal at the centre of the band.
The spectral density level represents the energy level in a band one cycle wide whereas by definition a tone has a bandwidth of zero.
There are two ways of transforming a signal from the time domain to the frequency domain. The first requires the use of band limited digital or analog filters.
The second requires the use of Fourier analysis where the time domain signal is transformed using a Fourier series. This is implemented in practice digitally (referred to as the DFT – discrete Fourier transform) using a very efficient algorithm known as the FFT (fast Fourier transform). Digital filtering is discussed in Appendix D.
1.11 COMBINING SOUND PRESSURES